Interactive proofs and a Shamir-like result for real number computations

Abstract

We introduce and study interactive proofs in the framework of real number computations as introduced by Blum, Shub, and Smale. Ivanov and de Rougemont started this line of research showing that an analogue of Shamir’s result holds in the real additive Blum–Shub–Smale model of computation when only Boolean messages can be exchanged. Here, we introduce interactive proofs in the full BSS model in which also multiplications can be performed and reals can be exchanged. The ultimate goal is to give a Shamir-like characterization of the real counterpart $${{\rm IP}_\mathbb{R}}$$ of classical IP. Whereas classically Shamir’s result implies IP = PSPACE = PAT = PAR, in our framework a major difficulty arises: In contrast to Turing complexity theory, the real number classes $${{\rm PAR}_\mathbb{R}}$$ and $${{\rm PAT}_\mathbb{R}}$$ differ and space resources considered separately are not meaningful. It is not obvious how to figure out whether at all $${{\rm IP}_\mathbb{R}}$$ is characterized by one of the above classes—and if so by which. We obtain two main results, an upper and a lower bound for the new class $${{\rm IP}_\mathbb{R}.}$$ As upper bound we establish $${{{\rm IP}_\mathbb{R}} \subseteq {\rm MA\exists}_\mathbb{R}}$$ , where $${{\rm MA} \exists_\mathbb{R}}$$ is a real complexity class introduced by Cucker and Briquel satisfying $${{\rm PAR}_\mathbb{R} \subsetneq {\rm MA}\exists_{\mathbb{R}} \subseteq {\rm PAT}_\mathbb{R}}$$ and conjectured to be different from $${{\rm PAT}_\mathbb{R}}$$ . We then complement this result and prove a non-trivial lower bound for $${{\rm IP}_\mathbb{R}}$$ . More precisely, we design interactive real protocols verifying function values for a large class of functions introduced by Koiran and Perifel and denoted by UniformVPSPACE $${^{0}.}$$ As a consequence, we show $${{\rm PAR}_\mathbb{R} \subseteq {\rm IP}_\mathbb{R}}$$ , which in particular implies co- $${{\rm NP}_\mathbb{R} \subseteq {\rm IP}_\mathbb{R}}$$ , and $${{\rm P}_\mathbb{R}^{Res} \subseteq {\rm IP}_\mathbb{R}}$$ , where Res denotes certain multivariate Resultant polynomials. Our proof techniques are guided by the question in how far Shamir’s classical proof can be used as well in the real number setting. Towards this aim results by Koiran and Perifel on UniformVPSPACE $${^{0}}$$ are extremely helpful.